Geometry of manifold of special holonomy and gauge theory/gravity correspondence

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D-branes in superstring theory provide a new method of investigating the mathematical structure of gauge theory. For example, D-branes in manifolds of exceptional holonomy group G_2,Spin(7) describe non-perturbative dynamics of supersymmetric gauge theory. In collaboration with Y.Yasui, we constructed explicit metrics of Spin(7) holonomy, that are of cohomogeneity one with SU(3)/U(1) principal orbit. In particular we found a new type of metrics that has asymptotically a circle S^1 of finite radius. This is a higher dimensional analog of four dimensional gravitational instantons ; Taub-NUT metic and Atiyah-Hitchin metic. We expect these metrics have some applications to M theory compactification.One of aims of string theory is a unification of gauge theory and gravity and gauge theory/gravity correspondence has been attracted much attention recently. When we consider topological gauge/string theory, the correspondence can be formulated mathematically more rigorous way. One of recent important achievements is instanton counting in four dimensional gauge theory by Nekrasov. In collaboration with T.Eguchi, we show that Nekrasov's partition function of instanton counting can be reproduced as a partition function of topological string theory. The partition function for SU(N) Seiberg-Witten theory is obtained by considering topological string whose target space is the ALE fibration of type A_{N-1} over P^1. Matter field can be incorporated by making a blow up on the target space. The methods used in this computation have close relations to various branches in mathematics, such as representation theory, combinatorics, knot theory and integrable systems.